Summary

In this example students examine and critique an argument which implies that it is not cost effective to pay for an automobile with increased fuel efficiency. Using a few reasonable assumptions shows that some of the writer's quantitative claims are not very accurate.

Learning Goals

Learning goals include

Analyzing quantitative arguments,

Devising reasonable assumptions

Analyzing effects of assumptions on costs and savings, and

Creating and using linear and exponential models.

Context for Use

This example can be used towards the end of a unit on compound interest.

Description and Teaching Materials

I. "Economists have discovered over the long run, a 20 percent increase in gasoline costs, for instance, will result in a 20 percent decline in gasoline consumption.

II."A recent report from the National Academy of Sciences, for instance, notes that the fuel efficiency of a large pickup could be increased from 18.1 miles per gallon to 26.7 miles per gallon at a cost to automakers of $1,466."

III. "It would take the typical driver 14 years before he would save enough in gasoline costs to pay for the mandated up-front expenditure [$1,466]."

IV. "A similar calculation for getting a large SUV up to 25.1 miles per gallon leads to a $1,348 expenditure and, similarly, more than a decade before buyers would break even."

V. "You could take that $1,466, for instance, put it in a checking account yielding 5 percent interest, and make a heck of a lot more money than you could by investing it in automobile fuel efficiency."

After reading the article, students are given the following question set:

Answer the following:

a. Which of the assertions can be checked without considerable research?

b. What assumptions would need to be made in checking assertion III?

c. What assumptions would need to be made in checking assertion IV?

d. What assumptions would need to be made in checking assertion V?

Answer the following:

a. Is the assertion III above reasonable? Explain why or why not.

b. What would be the effect of increased costs of gasoline on assertion III?

c. What would be the effect of increased miles driven per year on assertion III?

d. Assume the cost of gasoline in 2001 was $1.40 per gallon and that it would take 14 years for the "typical driver" to recover the $1466 through savings in gasoline costs. How many miles per year would the "typical driver" drive?

Answer the following:

a. Is the assertion IV above reasonable? Why or why not?

b. How would the savings be affected if the current MPG of large SUVs were lower than 18.1 MPG?

Answer the following:

a. Is assertion V above reasonable? Why or why not?

b. If the $1466 is placed in one account at 5% interest and the annual savings from gasoline are deposited in a second account earning 5% interest, compounded annually, how do the amounts in the two accounts compare?

Teaching Notes and Tips

Sample solutions:

Answer the following:

a. Which of the assertions can be checked without considerable research?

Assertions I and II (above) attribute quantitative findings to two authorities, the general group of "economists" and the National Academy of Sciences. Checking these two, especially the first one, would require considerable research. The first assertion does not enter into the latter assertions so it can be left alone. The latter assertions do use the results of assertion II, but it can be assumed while considering assertions III, IV, and V.

b. What assumptions would need to be made in checking assertion III?

To compute annual fuel costs, one needs to know the cost per gallon of fuel and the miles per year driven by the "typical driver."

c. What assumptions would need to be made in checking assertion IV?

In addition to the assumptions above (cost of gasoline and miles driven per year) one needs to know the current fuel efficiency (miles per gallon) of a "large SUV."

d. What assumptions would need to be made in checking assertion V?

In addition to the assumptions about assertion III (cost of gasoline and miles driven per year) one needs to know the following:

Are the savings from increased fuel efficiency also placed in the checking account? If so, how often are they deposited?

How frequently is interest compounded in the checking account?

How long is the money left in the checking account?

Answer the following:

a. Is the assertion III above reasonable? Explain why or why not.

First, we need to make assumptions about the cost of gasoline per gallon and the number of miles driven per year. The cost of gasoline when the article was written (2001) was approximately $1.40 per gallon. The number of miles driven per year by the "typical driver" is reported from different sources to be between 10,000 and 12,000 miles. Let's take 10,000 miles per year and $1.40 per gallon as our assumptions.

The cost of gasoline per year driving the less efficient (18.1 miles per gallon (MPG)) automobile is (10,000 miles X $1.40 per gallon)/18.1 miles per gallon = $773.48.

Assuming the automobile gets 26.7 miles per gallon yields a cost of (10,000 miles X $1.40 per gallon)/26.7 miles per gallon = $524.34.

So the savings (by modifying the automobile) is $773.48 - $524.34 = $249.14. Hence it would require $1466/$249.14 = 5.88 years to recover the $1466 through savings on gasoline costs. Note that we are assuming that the cost of gasoline remains constant at $1.40 per gallon. Of course, that did not happen – gasoline prices generally increase over time.

The article asserts that recovering the $1466 would require 14 years, quite different from the 5.88 years found above.

b. What would be the effect of increased costs of gasoline on assertion III?

Increased gasoline costs reduce the time required to recover the $1466.

c. What would be the effect of increased miles driven per year on assertion III?

If we increase the miles driven per year, we recover the $1466 sooner.

d. Assume the cost of gasoline in 2001 was $1.40 per gallon and that it would take 14 years for the "typical driver" to recover the $1466 through savings in gasoline costs. How many miles per year would the "typical driver" drive?

If we save $1466 in 14 years, then the average savings per year is $104.71 per year. In order to achieve this yearly savings, one would need to drive about 4,200 miles per year.

Answer the following:

a. Is the assertion IV above reasonable? Why or why not?

The assertion states: A similar calculation for getting a large SUV up to 25.1 miles per gallon leads to a $1,348 expenditure and, similarly, more than a decade before buyers would break even.

Let's make the same assumptions we made above about the cost of gasoline and the miles per year: $1.40 and 10,000 miles. We have to make one more assumption here: let's assume that the large SUV's fuel efficiency is 18.1 miles per gallon before any alteration. The situation is then analogous to the one above. The savings per years is $215.71 It would thus require 6.25 years to achieve the $1,348 in savings.

The article asserted that this would be more than a decade, or more than 10 years.

b. How would the savings be affected if the current MPG of large SUVs were lower than 18.1 MPG?

This would increase the annual savings on gasoline, therefore reducing the time required to recover the $1,348.

Answer the following:

a. Is assertion V above reasonable? Why or why not?

Let's make the same assumptions as above about cost of gasoline and miles driven per year: $1.40 per gallon and 10,000 miles.

We know from above that this yields a savings of $249.14 per year. The $1,466 at 5% interest earns $73.30 in the first year. Now if we leave the $1466 in the bank and assume that interest is compounded annually, after 10 years it amounts to: $2,387.96 but the savings on gasoline is 10($249.14) = $2,491.40. So, after 10 years, the assertion seems to be incorrect.

One could investigate these relationships further by graphing to two savings functions, one of which is linear and the other which is exponential:

B = 1466 X 1.05x - 1466, represents the savings earned by letting the money accumulate interest in the bank for x years and

b. If the $1466 is placed in one account at 5% interest and the annual savings from gasoline are deposited in a second account earning 5% interest, compounded annually, how do the amounts in the two accounts compare?

The amount in the first account after x years is B = 1466 X 1.05x.

Keeping our assumptions of 10,000 miles per year and $1.40 per gallon, the annual savings on gasoline is $249.14. Using a finite geometric series one can keep track of the amount, G, in the second account after x years:

G = 249.14(1 + 1.05 + 1.052 + ... + 1.05x-1) = 4982.80 X (1.05x - 1).

By graphing these two functions, one finds that the gasoline savings account G is larger after 7.14 years. For example at 10 years, the gasoline savings account contains $3133.66 while the $1466 account contains $2387.96. By 18 years the gasoline savings account is double the $1466 account.

Assessment

This is a lengthy and involved case study which invites a varied assessment strategy. A possible plan is outlined below:

i. Individual homework: Students read the article and come to class prepared to discuss their responses to the questions in part 1.

ii. Class discussion: Instructor facilitates a class discussion centered around part 1. At the end of the discussion, students should be in agreement on which assertions can be checked with a few basic assumptions. Students can also brainstorm about what some reasonable assumptions might be. The instructor may wish to point out that when this article was written, the price of gas was approximately $1.40 per gallon.

iii. Group work: Working in small groups, students should work through part 2.

iv. Class discussion: Instructor should facilitate the discussion centering on student solutions to part 2. Each of four groups could present their work on questions a, b, c, and d. Instructor should make sure students are comfortable with the types of calculations that arise.

v. Individual homework/assessment: Parts 3 and 4 make for nice individual assessment exercises. The questions in part 3 could be saved for an in-class test or quiz. The questions in part 4 make for a nice final individual assessment on compound interest and could be assigned for homework.

References and Resources

The article for this example appeared on national Review Online on August 14, 2001 and can be found on the Cato Institute's website: